Dynamics of the impact of COVID-19 on the economic activity of Peru

Background The COVID-19 virus impacts human health and the world economy, causing in Peru, more than 800 thousand infected and a strong recession expressed in a drop of -12% in its economic growth rate for 2020. In this context, the objective of the study is to analyze the dynamics of the short-term behavior of economic activity, as well as to explain the causal relationships in a Pandemic context based on the basic number of spread (Re) of COVID-19 per day. Methods An Autoregressive Distributed Lags (ARDL) model was used. Results A negative and statistically significant impact of the COVID-19 shock was found on the level of economic activity and a long-term Cointegration relationship with an error correction model (CEM), with the expected sign and statistically significant at 1%. Conclusion The Pandemic has behaved as a systemic shock of supply and aggregate demand at the macroeconomic level, which together have an impact on the recession or level of economic activity. The authors propose changing public health policy from an indiscriminate suppression strategy to a targeted, effective and intelligent mitigation strategy that minimizes the risk of human life costs and socioeconomic costs, in a context of uncertainty about the end of the Pandemic and complemented by economic, fiscal and monetary policies that mitigate the economic recession, considering the underlying structural characteristics of the Peruvian economy.

Theoretical, mathematical and econometric model Theoretical and mathematical model The assumptions In relation to the subsystem of the population facing a COVID-19 pandemic, the following assumptions are assumed following the methodology [1, 2] and adapted [3].
• The total population N is constant and is divided into 3 categories. i) the susceptible individuals (St) at the initial moment are the healthy population that can become infected.
ii) infectious individuals (It) can infect susceptible individuals. iii) recovered individuals (Rt), acquire immunity, and no longer have the transmission capacity.
• The birth and death rates do not vary. Immigration and emigration are not considered.
• At the start of the epidemic outbreak, only a small percentage of the population is infected Io = 1/N • At the initial moment, the number of susceptible and infected is positive.
• The interaction between S, I, and R individuals is that susceptible individuals become infectious and these become recovered or dead individuals.
About the economic subsystem, the following assumptions are made based on the aggregate supply and aggregate demand model [4]: • A short-term model, in which prices are rigid. The prices of goods and services are fixed.
• The components of aggregate demand (DA) are given by the consumption of goods and services (C), investments and services (I), public spending (G) and net exports of goods and services (XN). Then the aggregate demand remains: DA = C + I + G + XN.
Aggregate Demand is assumed to be affected by an exogenous disturbance λ. We consider the COVID-19 pandemic to be the exogenous shock or the shock that exogenously affects aggregate demand. Thus: DA = Y = DA (C, I, XN) + G + λ.
Consumption level C depends on price level p, real income level y, real interest rate i, real wage w, income tax rate t. It remains: C = C (p, y, i, w, t). Investment level I depends 2 on the price level p, on the real interest rate i, on the real price of the shares pa, on the real exchange rate e. It remains: I = I (p, y, i, pa, e). The level of public spending G is exogenous and determined by the Government. It remains: G = g. Net exports XN, depends on the price level p, real exchange rate e, real interest rate r, price of minerals and real energy pm *, real wage w and world income y *. It remains: XN = XN (p, e, i, w, y *, pm *).
• The Aggregate Supply (SA) depends on the price level p, real wage w, real interest rate r, real exchange rate e, climatic conditions c and technology τ. The COVID-19 pandemic is also considered to be an exogenous shock or the shock that exogenously affects aggregate supply. Thus Y = SA (p, w, i, e, c, τ) + λ

The structural form
The subsystem of the population facing a pandemic is reduced to 4 equations. Where, equation (1) represents the dynamics of individuals infected by the Pandemic.

̇= − … (1)
In equation (2) represents the change in the susceptible population: Equation (3) represents the change in the recovered population: The economic subsystem can be expressed in 6 equations that represent the aggregate supply and aggregate demand of the Peruvian economy: = It is the recovery rate, is the speed with which an individual goes from infected to recovered or the rate at which infected individuals recover or die. This rate represents the inverse of the average number of days that COVID-19 lasts, which is considered by experts to be 14 days.
From equation (10), R is cleared and the population subsystem facing the Pandemic can be reduced to a system of two equations, equation (1) and (2).
Assuming that S is constant and approximately equal to N under initial conditions. This assumption decouples equation (2), from equation (1), S = N and equation (2) remains as:

̇= − γ
Given: N = S, replacing ̇= − γ , is factored: The growth rate of the infection will be: From the differential equation (11), the solution or path for the infected population is derived: Where, r=β-ϒ. It is shown that the doubling time of infectious Td cases is given by: The epidemic ends as St falls below = , which is derived from the epidemic balance ̇= 0 (γ>β). In summary, the epidemic ends because of the lack of newly infected individuals and not because of the lack of susceptible individuals.
Theorem: If Re≤1, then It monotonically decreases to zero as t⇒∞. From this it is inferred: If Re> 1, the level of infected individuals It begins to increase, reaches its maximum, and then decreases to zero as t⇒∞. This scenario of increasing numbers of infected individuals is called an epidemic.
It follows that an infection can cause an epidemic in a susceptible population if Re> 1 or β> γ or β> κ * τ. Then the effective number Re is obtained as: = number of contacts of each individual per unit of time (day). This would affect the population that would be willing to work, increasing employment, production, and income in the economy, which would affect supply and aggregate demand negatively.
In the economic subsystem, replacing equation (2), (3) and (5)  Then, the function of the determinants of the level of economic activity could be estimated econometrically in equation (19). However, the availability of information suggests that we dispense with some operational variables such as real wages w, income tax t, level of international economic activity y*, and technology τ. For the price of the pa shares, the operational variable General Stock Index provided by the BCRP called igb is used. For climate c, the Lima temperature operational variable provided by SENANHI called tem is used. For Government expense g we 6 use the total number of samples provided by MINSA called prut. For international prices pm * we use the international price of the main mining export product that is copper as the operational variable, given by the BRCP called cob. We add a dummy variable associated with the application of the suppression strategy and subsequent gradual opening of economic activities implemented by the Government d1. Then, the causal relationships of economic activity in the short term would be: Equation (20) is specified in logarithms, it is then:

Econometric model
The Autoregressive Distributed Lags (ARDL) model analyzes time series, where the dependent and independent variables are related simultaneously and by lagged values [5][6][7]. The general ARDL model (p, q1, q2, q3 ... qk), is given by yt which is an endogenous variable, and x1, x2, x3... When the test statistic is above the upper critical value, the null hypothesis (Ho) is rejected and concludes that cointegration is possible and a CE model can be estimated. The series do not have a unit root, that is, they are stationary and it is achieved with the first difference, that is, they are integrated of order I (1).
Also, after the elaboration of the stylized facts, the Granger test of causality is considered [8].
That there is a correlation between two variables does not imply causality, that is, that one variable is correlated with another does not always imply that one of them is the cause of the changes in the values of another. Granger  index is calculated as a productive factor of 45% and 55% respectively.
As the COVID-19 operational variable, the basic propagation number (Re) is used. The estimated data of the Re indicator, are calculated from the SIR model methodology following the available methodology and adapted for the Peruvian case [3]. With this methodology, the 94 data were completed, making a regression and calculating Re according to the estimated parameters.
Therefore, estimates are made with a sample of 62 and 94, respectively, because the most recent sample is expected to have more significant results compared to the first month, which was more volatile and had more noise.
The operational variables General Stock Index of the Lima Stock Exchange, the data of the price at the end of the day is considered and the international price of copper is similar; the reference interest rate and the exchange rate are adjusted for inflation, the same as those provided by the